and
Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object $\mathbb{R}$ in there.
As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.
As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superscheme?s.
From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.
super ∞-groupoid, smooth super ∞-groupoid, super smooth ∞-groupoid
super-spacetime, super-Minkowski spacetime, super Poincare group
Some historically influential general considerations are in
Introductory lecture notes include
Yuri Manin, chapter 4 of Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988
L. Caston, R. Fioresi, Mathematical Foundations of Supersymmetry (arXiv:0710.5742)
Gennadi Sardanashvily, Lectures on supergeometry (arXiv:0910.0092)
The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in
Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)
Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48
and in
A summary/review is in the appendix of
Anatoly Konechny and Albert Schwarz,
On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)
Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486 A summary/review is in the appendix of
Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)
A review of all this as geometry in the topos over the category of superpoints is in
Formulation in terms of synthetic differential supergeometry is in
For many more references see at supermanifold.
Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in
especially in the contribution
The appendix there
means to sort out various sign conventions of relevance.
Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in
For more on this see at superalgebra.
Discussion related to G-structure and Killing spinors includes